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Third post in the ongoing series on important innovations in logic theory to be found in my works.
There is so much to say, I hardly know where to start. Perhaps I had best go back a little and draw your attention to the extensive work done in Future Logic in listing syllogisms.
Following Aristotle and his successors (notably his disciple Theophrastus on the 4th figure), with respect to actual categorical propositions (those of the form “S is P”), I have there listed and validated 19 primary moods and 25 secondary moods in the four figures. I point out incidentally that these 44 valid moods out of a conceivable number of 864 (i.e. a mere 5% validity rate) clearly demonstrate the need to study logic if we want to make sure we reason correctly.
Thereafter, after discussing the various types and categories of modality, I did a similar treatment with respect to modal categorical propositions – meaning primarily for natural modality. Aristotle (and others) had indeed done considerable work in this field, but as I have pointed out in a previous blog, his understanding of modality there (though not in his philosophical discussions, note well) seems to have been limited to the logical mode of modality (or perhaps, I sometimes speculate, some sort of generic mode, underlying all the others) – and this led him to make some serious errors in his list of modal syllogisms.
Moreover, Aristotle’s listing of modal syllogisms was not as thorough and systematic as his listing of actual syllogisms; and so far as I know no one after him has managed an exhaustive and reasoned listing. In Future Logic, I do just that. I begin, as already said, by analyzing and explaining the main varieties of modality. Then, focusing on the crucial “de re” modes (as against the “de dicta” or logical mode, which is the usual object of study of modern logicians), I develop a full list of propositions, examine their oppositions and eductions (i.e. immediate inferences) and then their deductions (i.e. syllogisms, mediate inferences).
One novelty that greatly facilitated my inventory of the various possibilities was the introduction of the symbols n, a, and p for natural necessity, actuality and potentiality, respectively – which notation could be used either independently or as suffices to the traditional A, E, I, O symbols (for plural propositions – to which I added R and G for singulars). Another novelty in this context was to formulate general principles for quantification and modalization of oppositions as well as for intermodal oppositions – expressed in rectangular and three-dimensional ‘figures of opposition’ derived from the traditional ‘square of opposition’.
Finally, I systematically develop a list of valid modal syllogisms, including tables with the valid modes of quantity (all, some, this one), polarity (i.e. positive or negative – what is traditionally but misleadingly called ‘quality’), and modality (mainly of the natural type, and by analogy of the temporal type). The results obtained are sobering. Out of a total 108,000 theoretical combinations of modal and non-modal premises and conclusions, including mixtures of natural and temporal modalities, and including the earlier mentioned wholly actual syllogisms, only 1486 (1.4%) were found valid – and of these, only 93 (6.3%) were primary and the rest (93.7%) were secondary (i.e. implied in the main 93, and relatively less used though not never used).
Validations were of course done using the traditional methods of exposition, direct reduction and reduction ad absurdum (indirect reduction), as appropriate to each figure and mood. The results obtained for modal syllogism demonstrate again that reasoning cannot be left to “instinct” but requires serious investigation – not only to avoid invalid forms of reasoning, but to be made aware of valid forms that are not immediately obvious. The practical value of such knowledge is incalculable. Many more discoveries and insights are to be found in these chapters.